Math 5 Unit 1 Lesson 10 Multiplication Museums In 1907 three brothers from a farm in central Alberta found a way to make history in Canada. They heard about the Wright Brothers creating the first flying machine so they decided to make their own. John Underwood was the first man in Canada to be lifted into the air by an aircraft. The Alberta Aviation Museum has an exhibit that details the contributions of the Underwood brothers to the history of aviation in Alberta. These early pioneers and others can be studied there along with many aircraft. The 5th grade class at Mountain Spring Elementary School is going on a field trip to the Alberta Aviation Museum. They have 58 students that will go on the trip. The admission for each student is $5. To find the amount they will pay for the students to enter the museum you need to multiply 58 students by 5 dollars. Math 5 1-105 Math 5 Unit 1 Lesson 10: Multiplication Reflection How do you multiply a two-digit number by a one-digit number? Objectives for this Lesson In this lesson you will explore the following concepts: • Multiply 2-digit by 2-digit numbers • Solve 2-digit multiplication problems Go online to complete the Concept Capsule: Understanding Distributive Property Using Base 10 Blocks. Multiplying 3-Digit Numbers by 1-Digit Numbers You should remember that multiplying three digits by a number from 1 through 9 may be done using many methods. Here is a quick review of a couple of those methods that may help you as we move on to more challenging problems. Distributive Property The distributive property makes multiplying large numbers a bit easier for you. It involves breaking down your two or three digit number into expanded notation based on place value. This helps you to keep the value of each place as it should be. The concept is the same for both examples. There are two different ways to organize your work. 1-106 Math 5 Unit 1 Lesson 10: Multiplication Example 1 Multiply 234 x 8 Write 234 in expanded notation: 200 + 30 + 4 Multiply each term by 8: (200 x 8) + (30 x 8) + (4 x 8) 1 600 + 240 + 32 Add each product: 1 872 Example 2 Multiply 315 x 6 Line up the ones digits to multiply: 315 6 30 60 + 1 800 1 890 x 6x5 6 x 10 6 x 300 315 x 6 = 1 890 Notice that the second example uses the same concept as the first. The difference is how it is organized. Multiplication by Carrying Another way to multiply is by using an old method in which you carry over place values. You simply line up the place values of the numbers and multiply going down. Math 5 1-107 Math 5 Unit 1 Lesson 10: Multiplication Example 3 Multiply 24 x 3 Remember, line up the ones and use basic facts. First: Multiply 3 x 4. Since the answer is 12 you will need to carry the 1 to above the tens digit. 1 24 x3 2 Next: Multiply 3 times the 2 in the tens digit. You will get 6 but you must ADD the 1 that you carried. 1 24 x3 72 24 x 3 = 72 Now It’s Your Turn Multiply. Use estimation to check. a. 24 x 5 b. 87 x 4 c. 124 x 6 Solutions a. 120 b. 348 c. 744 Multiplying 2-Digits by 2-Digits This goes way beyond multiplying by one digit. The distributive method using place value is a very important part of any multiplication problem. 1-108 Math 5 Unit 1 Lesson 10: Multiplication Let’s Explore Exploration 1: Using Array Models Materials: Unit 1, Lesson 10, Exploration 1 page from your Workbook, Pencil The problem 18 x 12 can be modeled in this manner: 18 = 10 + 8 10 8 10 x 10 8 x 10 10 10 x 2 8x2 2 12 = 10 + 2 This models: 18 x 12 = (10 x 10) + (8 x 10) + (10 x 2) + (8 x 2) Multiply the problem in each box: 10 8 100 80 10 20 16 2 Add the numbers in each box to get the product: 100 + 80 + 20 + 16 = 216 18 x 12 = 216 For 1 – 3: Write the distributive property for the multiplication model. 1. ___________ + ___________ + ___________ + ___________ 20 x 10 6 x 10 20 x 3 6x3 Math 5 1-109 Math 5 Unit 1 Lesson 10: Multiplication 2. ___________ + ___________ + ___________ + ___________ 30 x 20 5 x 20 30 x 4 5x4 3. ___________ + ___________ + ___________ + ___________ 80 x 40 7 x 40 80 x 3 7x3 For 4 – 5: Model each product and find the answer. 4. 54 x 48 5. 74 x 32 6. Write your own multiplication problem of two digit numbers. Model the problem and find the answer. Using the Distributive Property The distributive property can be used to multiply two digit numbers. Here is an example. Example 4 Multiply 54 x 27 and check by estimation. Another way of organizing: 1. M ultiply the ones and write your answer down. 2. M ultiply the ones from the bottom number to the tens of the top and write your answer down. 7 x 50 20 x 50 54 x 27 28 350 80 1 000 1 458 3. M ultiply the tens from the bottom number to the ones of the top and write your answer down. 4. Multiply the tens and write your answer down. 1-110 7x4 20 x 4 Math 5 Unit 1 Lesson 10: Multiplication Check: Estimate the solution using compatible numbers: 50 x 30 = 1 500. Decide if your answer is reasonable by comparing it to the estimate. 54 x 27 = 1 458 You can also use the carrying method to multiply two digit numbers. Example 5 Multiply 25 x 14 25 x 14 Line up the place values to begin: Multiply the ones place of each number. In this case 4 x 5: Carry the 2 from 5 x 4 = 20 2 25 x 14 0 The 0 from 5 x 4 = 20 goes here ultiply the ones of the second number by the tens of the first M number. In this case 4 x 2: 2 After 4 x 2 = 8 ADD the 2 you carried 25 x 14 100 Cross out the 2 once you have added it. Multiply the tens of the second number by the ones of the first: 2 1 x 5 goes in front of place holder 25 x 14 100 50 0 is a place holder The place holder holds the ones place because here you are actually multiplying 10 x 5 since the value of the 1 in 14 is 10. Math 5 1-111 Math 5 Unit 1 Lesson 10: Multiplication Multiply 1 times 2 – put in front of the 50: 2 25 x 14 100 250 2 25 Add 100 + 250: x 14 100 + 250 350 This method all depends on keeping place values of each number in the same column. You could turn a piece of notebook paper sideways and use the columns rather than the lines to work your problem: hundreds tens ones 2 25 x 14 100 250 350 This will keep all of the place values lined up and help you organize your work. Example 6 Multiply 86 x 42. Estimate to check your solution. 1-112 21 86 x 42 172 3 440 3 612 Line it up and multiply: heck: If you estimate, the solution is 90 x 40 = 3 600 using compatible C numbers. The answer is reasonable since it is so close to the estimate. 86 x 42 = 3 612 Math 5 Unit 1 Lesson 10: Multiplication Another way to look at this method: 2 x 86 = 172 and 40 x 86 = 3 440. These are the two numbers you get when you use the carrying method. You are really using distributive property: 86 x (40 + 2). So no matter what method you use it all goes back to that concept. Practice these methods to determine which one works for you. The main thing is to be comfortable with the method you are using and to eliminate careless errors. Now It’s Your Turn Multiply. Estimate to check the solution. a. 35 x 12 b. 49 x 11 c. 67 x 12 Solution a. 420 b. 539 c. 804 Lattice Multiplication Another method for multiplying is called lattice multiplication. If you remember pictures better than words, you will like this method. Before you begin you need to practice writing answers to simple problems in a different form. Math 5 1-113 Math 5 Unit 1 Lesson 10: Multiplication Write 2 x 9 = 18 in lattice form: 2 1 9 8 Notice that the 2 and the 9 are outside the box. The 1 for the tens place is in the top part of the divided box. The 8 for the ones is in the bottom. 3 x 7 = 214 x 5 = 201 x 7 = 7 3 4 2 1 2 1 7 0 5 0 7 7 Notice the 0 in the tens for 1 x 7. That will be important. Now, how does this apply to 2-digit by 2-digit multiplication? Here is an example: Example 7 Multiply 42 x 15 1. Set up your lattice with the 42 along the top and 15 down the right side: 2. Multiply each part as shown: 4 2 0 4 4 1 5 1-114 2 0 4 0 4 2 1 5 4 2 0 0 4 2 0 2 1 5 2 0 0 4 2 2 1 0 0 1 5 Math 5 Unit 1 Lesson 10: Multiplication Notice that the products are being placed as we did with our smaller ones earlier. The tens digits are on top of the diagonal and the ones digits are on the bottom. 3. Now take the final lattice and add the diagonals in this order: Diagonal 1 4 Diagonal 2 2 0 4 0 4 2 1 2 2 2 0+4+2=6 0 3 0 5 0 4 2 1 1 0 Diagonal 4 0 4 2 1 2+1+0=3 2 0 1 0 Diagonal 3 4 0 4 0 0 0 1 5 0 2 5 0 0 2 0 4 0 1 1 2 0 6 2 3 0 5 0 4.Now simply gather the digits from top left side to bottom right side of your box and THAT is the answer! 4 2 0 0 1 0 4 2 1 2 5 6 0 0 3 Math 5 42 x 15 = 630 Leave off the 0 that is in the first position. 0 1-115 Math 5 Unit 1 Lesson 10: Multiplication Example 8 Multiply 47 x 53. Estimate to check your answer. Estimate using compensation first: 50 x 50 = 2 500 Using lattice multiplication: 4 2 4 2 3 0 1 5 2 2 9 7 1 5 3 1 Based on our estimate of 2 500, our answer 2 491 is reasonable. 47 x 53 = 2 491 What if the sum of a diagonal is larger than 9? Simply carry the tens digit to the next diagonal to the left. Example 9 Multiply 84 x 62. Estimate your solutions. An estimate is 80 x 60 = 4 800 5 1 + 2 + 8 + 1 = 12 2 4 + 0 + 6 = 10 1-116 4 1 2 8 1 4 8 1 4 0 6 0 8 6 2 8 Notice that the 1s were carried to the next diagonal. 84 x 62 = 5 208 Math 5 Unit 1 Lesson 10: Multiplication Let’s Practice • In your Workbook go to Unit 1, Lesson 10 and complete 1 to 25. Problem Solving Once you know how to multiply two digit numbers you are ready to solve problems involving multiplication. Here are some clue words you can look for in problems involving multiplication: in all per product of times each every Example 10 Write a number sentence for the following problem. Solve. Alyssa and Zach are planting trees for an environmental service project. They planted 12 rows of trees with 18 trees in each row. How many trees did they plant in all? What is the question? How many trees did they plant? What do you know? They planted 12 rows with 18 in each. Write a number sentence: 12 x 18 Solve 1 18 x 12 36 + 180 216 Check: Estimate by compatible numbers 20 x 10 = 200 so 216 is reasonable. State the solution: Alyssa and Zach planted 216 trees. Math 5 1-117 Math 5 Unit 1 Lesson 10: Multiplication Example 11 Write a number sentence for the following problem. Solve. The new middle school has 84 classrooms. There are 24 brand new desks per classroom. How many students in total can be seated in the new classrooms at one time? What is the question? How many seats are in the classrooms? What do you know? 24 seats per room for 84 classrooms. Write a number sentence: 84 x 24 Solve 84 24 16 320 80 + 1 600 2 016 x 1-118 4x4 4 x 80 20 x 4 20 x 80 Check: Estimate by compatible numbers 80 x 20 = 1 600 State the solution: 2 016 students can be seated in the new classrooms at one time. Math 5 Unit 1 Lesson 10: Multiplication Let’s Practice • In your Workbook go to Unit 1, Lesson 10 and complete 26 to 31. Go online to watch the Notepad Tutor: Multiplication with Decimals (1-Digit Whole Number Multiplier). Math 5 1-119 Math 5 Unit 1 1-120 Lesson 10: Multiplication

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